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About Heisenberg Uncertainty Relation
arXiv:quant-ph/9903100 v2 15 Jun 2000
By E. Schrödinger
Proceedings of The Prussian Academy of Sciences
Physics-Mathematical Section. 1930. XIX, pp.296-303 +
Abstract
The original Schrödinger’s paper is translated and annotated in honour of the
anniversary of his Uncertainty Relation [Bulg.J.Phys.,vol.26,nos.5/6 (1999)
pp.193-203]. In the annotation it is shown that the Uncertainty Relation can be
written in a complete compact canonical form.
70−th
ANNOTATION
by A. Angelow
++ ,
M.-C. Batoni
+++
The main reason to publish the original Schrödinger’s paper in English, is the fact that no
one of the books on Quantum Mechanics cites it (see for example [1† -15† ]). Actually, the
Schrödinger’s paper is chiefly based on the notes of the seminars of Physics-Mathematical
Section of The Prussian Academy, where many famous physicists worked to establish the
underlying basis of Quantum Theory. Being a kind of internal report, this work remained,
for many years at a certain marginal distance from the physicist scientific awareness.
Another argument in favour of its oblivion concerns the enthusiastic discussions, mostly
about the physical interpretation of the uncertainty principle, rather than its mathematical
straightforward derivation. After Schrödinger, the very first appearance of the new uncertainty relation occurs in the book of Merzbacher. However, he has not pay any attention to
the new term (– the covariance) and directly derives the Heisenberg inequality [11† ]. The
same embarrassment appeared in [12† ] and [13† ].
Fortunately, over the last years the scenario changed: for example, in the field of Quantum
Optics, which was avocated to demonstrate the fundamental limit of Quantum Theory with
understanding underlying on that new term, two monographs [14† ,15† ] were published, but
the authors missed to cite the original Schrödinger’s paper.
Schrödinger’s work, originally written in German, was translated only in russian by A.
Rogali [16† ], in 1976. However, we would like to emphasize that the sentence (*), written
in the eleventh line after equation (11), is wrongly translated and it contradicts to the
whole paragraph [16† ]. In that paragraph Schrödinger established classes of states with
non-vanishing covariance.
Finally, we would like to demonstrate the essential contribution of Schrödinger’s inequality presenting it in a new compact form, using a modern terminology from mathematics,
rather than that used in 1930. Let us take the three independent second-order central moments of the joint quantum distribution of two variables A and B, which are of special
interest and warrant a special notation (see, for ex. [17† ,18† ]):
E[(A − E[A])2 ] ≡ (A − A)2 ≡ V ar[A] ≡ ∆(A) 2
(1† )
E[(B − E[B])2 ] ≡ (B − B)2 ≡ V ar[B] ≡ ∆(B) 2
(2† )
1
(A − E[A])(B − E[B]) + (B − E[B])(A − E[A])
≡ Cov[A, B] ≡ ∆(A, B)
E
2
(3† )
where E[.] means the expectation value. Obviously, the fourth second-order moment is
Cov[B, A] = Cov[A, B], (respectively ∆(B, A) = ∆(A, B)). Note that when the observables
X and Y don’t commute, the correct expression for their product is not XY , but the
X
. We conclude that the covariance is equivalent to the new term
symmetrized one XY +Y
2
in the Schrödinger’s inequality:
Cov[A, B] =
AB + BA
−AB
2
(4† )
Let us construct the so-called [19† ] covariance matrix (keeping in mind the non-commutativity of the observables in contrast to [19† ]):
σ[A, B] ≡
V ar[A]
Cov[B, A]
Cov[A, B]
V ar[B]
≡
∆(A) 2
∆(B, A)
∆(A, B)
∆(B) 2
(5† )
Now, we can write the Schrödinger Uncertainty Principle ( see below-eq.(9) ) in the canonical form
det(σ[A, B]) ≥
1
2
|[A, B] | ,
4
or , for position and momentum
det(σ[q, p]) ≥
h̄2
,
4
(6† )
and it is easy to see that the uncertainty relation is invariant under the rotation transformation in the phase space, while the Heisenberg one is not. We would like to emphasize
that the new term in the inequality also plays an important role in the method of linear
invariants in Quantum Mechanics, where the covariance is expressed in terms of the solution of the equation of a non-stationary two-dimensional harmonic oscilator [20† ].
Translated and annotated in honour of 70−th anniversary of Schrödinger Uncertainty Relation.
Sofia, January 1999
São Paulo, March 1999.
PACS: 03.65.-w
§1. Recently E. U. Condon and H. P. Robertson [1] took into consideration the generalization of the fundamental principle of the quantum mechanics - that of the uncertainty
- over an arbitrary canonical non-conjugate couple of physical variables. Trying to reach
the same, I arrived at a slightly wider generalization than the Robertson’s one, which is,
in fact stronger than the original Heisenberg inequality.
First of all, let us set out what is well known. The state-of-the-art of the “interpretation
question” is the following: the test domain is a single specific physical system. The base
for the system knowledge that we dispose of - the catalogue of all that we can assert
about the system - is equivalent to a complex function Ψ in the coordinate space of the
system (it changes in a regular manner in time, but is not important at the moment).
The mathematical correlate of a “physical variable”, i.e. of a very specific measurement
that one might apply to the system, is a very specific linear Hermitean operator that from
each Ψ-function produces an other such a Ψ-function. One can calculate the expectation
2
value of the respective measurement from the measure operator, say A, and the given
Ψ-function:
Z
A = Ψ∗ AΨdx
(1)
(Ψ∗ is the complex conjugate, the integration
goes over the whole coordinate space; given
R ∗
that Ψ is constantly normalized, i. e. Ψ Ψdx = 1).
The meaning of the expectation value is: a mean value by an unlimited number of
measurements, while one must be sure that the system state is the same before each
measurement, not changed by the measurement itself. In general, all possible statements
one can make about the system are encoded in the expectation values. Moreover, one
should keep in mind that it is up to us to choose the marking of reference scale of our
measurement instrument. We can, for example, set a value one only to one scale division
and zero to all other. There is a specific operator attached to this “measurement” – one
could name it as an operator in blinkers†† , V.Neumann named it identity operator. The
respective expectation is obviously nothing else but the probability of the corresponding
measurement value or measurement value intervals. The Ψ-function determines also the
total measurement statistics.
The average error or the mean uncertainty of the value, which belongs to the operator
A, is defined as
q
q
∆A =
(A − A)2 =
A2 − (A)2
(2)
(where in the first of the two expressions A should be more precise: A multiplied by the
identity operator.) It may be proven, that this definition is not only formally constructed
according to the theory of errors, but ∆A is really the average error of the variable A,
when the statistics is defined in the above given way.
To prove now, that the product of the uncertainties of two random variables A and B
satisfies the Heisenberg or even more precise inequality, we need to denote the following
mathematical statements:
1. the Hermitean character of A implies that the expectation value (1) is constantly
real;
2. for each Hermitean operator it holds
Z
f Agdx =
Z
gA∗ f dx,
(3)
i.e., it could be rolled over on the other factor in such an integral, in this case the operator
transforms into its conjugate form [2];
3. the product of two Hermitean operators is in general not Hermitean, but it could
be split into a “symmetrical product” and half of its commutator:
AB =
AB + BA AB − BA
+
2
2
(4)
The first term is Hermitean,
the last one is “skew Hermitean”, i.e. it becomes Hermitean
√
multiplied by i = −1. The splitting in many aspects corresponds to the splitting of a
random (complex)† number into real and imaginary parts. Immediately from here one
might extract the splitting of the expectation value into real and imaginary parts. The
“expectation value” of every commutator is pure imaginary.
3
4. Finally, we need the so-called Schwartz inequality [3]
(a1 a∗1 + a2 a∗2 + ... + an a∗n )(b1 b∗1 + b2 b∗2 + ... + bn b∗n ) ≥ |a1 b1 + a2 b2 + ... + an bn |2 ,
(5)
that we shall apply in a limiting case on the continuous range of the values of both
functions f and g in the coordinate space:
Z
∗
f f dx ·
Z
Z
∗
gg dx ≥
We assume here that specially
2
f gdx .
(5′ )
g = A∗ Ψ∗ ,
f = BΨ
(6)
where A and B are some Hermitean operators and Ψ is an arbitrary wave function, i.e. an
arbitrary continuous and normalized function in the coordinate space. Using the equation
(3) one obtains
Z
∗
2
Ψ B Ψdx ·
Z
∗
2
Ψ A Ψdx ≥
i.e., in terms of the notation (1)
Z
Ψ
∗
2
ABΨdx ,
(7)
2
(7′ )
A2 · B 2 ≥ AB .
If we decompose the right hand side according to (4), then we get
A2
·
B2
≥
AB + BA
2
!2
AB
+
2
− BA .
2
(8)
This is already the inequality that we need to proof, but only in the special case when
A and B vanish. In order to arrive at the general case, one should apply (8) and instead
of the operators A and B rather use the following
A − α1 and B − β1.
First of all α and β must be arbitrary real constants, α1 is the (identity)† operator
multiplied by α. The resulting inequality is therefore valid: 1. for an arbitrary Ψ, 2.
for every real pair of constants α, β. Therefore, there is no limitation on the Ψ-function
to influence the choice of the pair of constants and especially to set
β = B.
α = A,
Finally, we end up with:
2
2
(∆A) (∆B) ≥
AB + BA
−AB
2
!2
AB
+
2
− BA .
2
(9)
This is the final form. The first from the two addends on the right hand side is a new one
(to the best of my knowledge). (Without that term the inequality stands as the one of H.
P. Robertson.) So, the inequality links together three quantities: 1. the product of the
mean deviations squared, 2. the absolute value squared of half of the mean value of the
4
commutator, 3. a quantity which could be defined as a square of the mean deviationsproduct (the covariance)† in the condition that non-commutability is taken into account,
i.e. the mean deviations-product must be define as the arithmetic mean of
(A − A)(B − B) and (B − B)(A − A)
(10)
which are the “mixed” expressions ( ≡ covariances, see eq.(3† ) )† , completely analogous
to (∆A)2 and (∆B)2 ††† .
One is led to the Heisenberg inequality when the last mentioned quantities are stricken
out in order to make stronger the inequality and A, B are chosen to be canonically
conjugate:
h
.
AB − BA =
2πi
Then it results in
h
∆A · ∆B ≥
.
(11)
4π
On the other hand, it is known that the Heisenberg limit is not really too low, but for
some special Ψ-functions achieves even higher value [4]. This implies that at least for
these special Ψ-functions the (central)† mean deviations-product of the canonical conjugate operators vanishes. This will be used in §2.
In the classical theory of errors or fluctuation theory it is well known that the vanishing
of the mean deviations-product is a necessary (but not sufficient) condition for two values
to fluctuate totally independent one from another. While canonically conjugate quantum
variables have some “independence” that could mean that some precise knowledge about
one excludes such a knowledge about the other, so one could perhaps suppose that their
mean deviations-product, i. e. for each Ψ-function, has vanishing expectation value. But
this is not the case(∗ ). Let us consider the two canonically conjugate operators
A=x
B=
h ∂
,
2πi ∂x
so we get [5]
1
2πi AB + BA
=
h
2
2
Z
Ψ
∗
"
#
∂Ψ
∂
1
x
+
(xΨ) dx =
∂x
∂x
2
Z
∂Ψ∗
x Ψ
dx
−Ψ
∂x
∂x
∗ ∂Ψ
!
1Z
∂
Ψ
=
xΨ∗ Ψ
ln ∗ dx
2
∂x
Ψ
A=
2πi
B=
h
Z
xΨ∗ Ψdx
Z
∗ ∂Ψ
1
Ψ
dx =
∂x
2
Z
∂Ψ∗
dx
Ψ
−Ψ
∂x
∂x
∗ ∂Ψ
!
∂
Ψ
1
ΨΨ∗
ln ∗ dx.
=
2
∂x
Ψ
iΦ
Let now Ψ = re with real Φ and real, non-negative r, which must satisfy the normalizing
condition
Z
r 2 dx = 1
Z
5
Then we get:
2π
h
!
AB + BA
− AB =
2
Z
xr
2 ∂Φ
∂x
dx −
Z
2
xr dx ·
Z
r2
∂Φ
dx.
∂x
is any real function and r 2 is an absolute non-negative function (not taking into
As ∂Φ
∂x
account the normalizing condition), so we get that in general the right hand side does
not vanish. One needs for example to choose r 2 to be even and ∂Φ
to be odd (and not
∂x
identically vanishing), so the deviation product is surely positive.
As is known, the canonically conjugate quantum variable is not unambiguously defined. If B is conjugate to A this implies that B + εA is as well (ε is any real number).
With this change the mean deviations-product changes too, and becomes, as one could
easily calculate, ε(∆A)2 . In the same manner, the result will be ε(∆B)2 , if A was changed
to A + εB. This can always make the deviations-product equal to zero by changing one
of the operators, without changing their canonical relation. The change depends on the
above shown special Ψ-function of course. One can not reach an identical vanishing of
the deviations-product in such a manner.
§2. To the discovery of the complete inequality (9) we are led, by a chance, to the
following question, which is interesting by itself. Let us consider a force free mass point,
p2
mass m, coordinate q, momentum p, Hamilton-function H = 2m
. I must undertake
simultaneous measurements of the coordinate and momentum at “time zero”, with highest
possible precision, i.e. so that
h
.
(12)
∆q0 ∆p0 =
4π
Further I could distribute the error on q0 and p0 so that for a given later time point t,
could achieve the most precise place. This means ∆q to become the least possible. We
use for this purpose the very convenient “q-number-method”, which is in a methodical
manner opposing to the wave mechanics. I would like to elucidate shortly on it here,
repeting what is well known. For the theorist working on a wave mechanics the operator,
which corresponds to a specific physical variable, does not change in time. If one wants to
know the mathematical expectation value for this variable, one calculates the Ψ-function
for this later moment from the “time-dependent wave equation”. Then one applies the
corresponding operator, which mentioned above, which is the same for every moment. On
the other side the q-number-theorist has to operate with one single Ψ-function at one single chosen moment, once and for all. However it is unnecessary to express any statement
for it, once the moment is totally arbitrary chosen. One assumes, that the operators are
time dependent and we may ask instead: how does the operator change itself in time, i.e.
which operator should be applied on the original Ψ-function, in order to calculate the
mathematical expectation of the respective value at the time t?
Here we point out, that one may calculate the operators (or q-numbers, or matrices)
almost as the usual numbers, and indeed, their change in time is determined by the equation of motion of the classical mechanics. The only difference is that, occasionally,
when it is the case, one should pay special attention to an eventual non-commutability of
the operator multiplication.
6
So, in this present simple case, the integration of the equation of motion reads:
t
p0 .
m
q = q0 +
One can directly make from it the mean square deviation of the coordinate, (∆q)2 , for
every moment t:
!2
2
t
t
(∆q)2 = q0 + p0 − q0 + p0
m
m
2t
= (∆q0 ) +
m
q0 p0 + p0 q0
− q0 p0
2
2
!2
+
t2
(∆p0 )2 .
2
m
The middle term above is essentially the mean deviations-product of q0 and p0 , which
vanishes, in accordance with the prediction, when q0 and p0 are determined with optimal
precision. Then we simply have
(∆q)2 = (∆q0 )2 +
t2
(∆p0 )2
m2
or using (12)
h
(∆q) = (∆q0 ) +
4π
2
2
!2
t2
1
,
m2 (∆q0 )2
see Ref.[6].
This expression becomes a minimum for that value of (∆q0 )2 , which makes both addends
on the right hand side equal, i.e. for
ht
;
4πm
(∆q0 )2 =
(∆q)2 is then exactly twice the value of (∆q0 )2 , i. e.
∆q =
s
ht
.
2πm
(13)
It seems to me, that this final result is likely to have two points of interest. First,
the proportional relation with the square root from the time, which makes allusion to
well known classical deviation principles. Secondly, that the statement has an remarkable
absolute character, namely, the precision attainable in a later moment depends only on
intermediate time and not on the initial momentum. For a free electron, as an example,
one might give a place prognosis for the end of the first second on the bases of already
taken measurements of position and momentum, in the most favorable case with a precision of 1cm, quite independent of whether the electron is fast or slow [7].
Of course, at a very high speed this will be changed as it should take into consideration
the relativity theory. I believe that this could occur by the following simple considerations.
The equation (13) is applied to the rest reference system of a point mass. Let m be the
rest mass, tr , will be the internal time, so
(∆q)r =
s
7
htr
.
2πmr
(14)
This is the precision that is attainable for a moving observer when calculating the position
of the point mass of the co-moving system for the moment, called “tr seconds later”. When
the observer shows his knowledge through signs in the√space, to the “rest observer” those
1 − β 2 : 1 ; further he ought say
signs seem to be nearer to each other in relation of
looking from his point of view that the prognosis were made for the time interval
t= √
tr
,
1 − β2
(15)
because for him the clock, with which all the statements of the moving observer were
made, seemed run slower than his own clock. From his point of view, the mean error
decreases
v
s
u √
q
q
u ht 1 − β 2
ht
t
∆q = 1 − β 2
= 1 − β2
.
(16)
2πmr
2πm
It becomes smaller and comes nearer to zero when the velocity approaches are nearing
the speed of the light. This happens not only when the mass m goes to infinity, but also
for a series of point masses moving with an ever growing speed and an ever smaller rest
mass such that all the moving masses m keep the same value m. Even in this case the
maximum precision grows unlimited with the velocity approaching the speed of the light.
This is indeed satisfying, since this is a boundary process that gives a hope to obtain
an accurate statement for a light quantum. And this is really true for light quantum
because the Maxwell waves exhibit no dispersion, and preserve indefinitely long the place
precision which they got in the beginning, and it could indefinitely grow, since the strong
momentum dispersion which is connected with it, does not have bad influence.
Reported on the 5th June 1930
Joint-Staff Meeting on the 19th June 1930
Distributed on the 16th July 1930
References
[1] E. U. Condon, Science, 31. Mai (1929);
H. P. Robertson, Phys. Rev. 34 (1929) 163.
Mr. A. Sommerfeld was so kind to point out to me these two notes, when I told him
about the following considerations.
[2] A∗ is so defined that evidently A∗ Ψ∗ = (AΨ)∗ , and therefore A∗ Ψ = (AΨ∗ )∗ .
[3] Vgl. H. Weyl, Gruppentheorie und Quantenmechanik, Hirzel, Leipzig, (1928) p.272.
This proof is closely connected to the proof of the Heisenberg inequality given there.
[4] J.v.Neumann, ZS.f. Phys. 57 (1929) 34;
W. Heisenberg, Ibid. 43 (1927) 187;
G.G. Darwin, Proc. Ray. Soc. A 117 (1927) 268.
[5] All integrals are from −∞ to +∞.
8
[6] This equation has already been developed by Heisenberg in his first work on the
uncertainty principle (Zeitschr. f. Phys. 43 (1927) p.188), but for the special case of
pseudo arbitrary character Ψ-function.
[7] Of course, there are wave functions that, in a precise given later moment, can determine the position with an arbitrary given precision. But one needs simply the chance
to be able to trace back such an approximation of a “maximal function” by means
of a wave equation over the corresponding time interval, and the resulting function
to be taken as original state. But it should be not less than “optimal”.
———————–
) – Note added from the translators.
††
) – By this Schrödinger means that the operator does not change the direction and modulus of the vector, as the horse in blinkers does not change the direction and the speed
until this is not required by the driver.
†††
) – Indeed, if one put B = A in (3† ), then Cov[A, A] = V ar[A] ≡ ∆(A)2 .
†
+
) Reprinted in: ERWIN SCHRÖDINGER, Gesammelte Abhandlungen, Band 3, Wien,
Verlag der Österreichischen Akademie der Wissenschaften, pp.348-356 (1984)
++
) Permanent address:
Institute of Solid State Physics, Bulgarian Academy of Sciences, 72 Trackia Blvd., 1784
Sofia, Bulgaria.
+++
) Permanent address:
Instituto de Fisica Teorica, Universidade Estadual Paulista, Rua Pamplona, 145, CEP
01405-900 Bela Vista, São Paulo, Brazil.
References
added from translators.
[1† ] W.Heisenberg, The physical principles of the quantum theory, New York, Dover,
(1930);
[2† ] J. Von Neumann, Mathematical foundations of quantum mechanics, Princeton,
NJ, University Press, (1955);
[3† ] P.A.M. Dirac, The principles of quantum mechanics. 4th rev. ed., Oxford, Clarendon Press (1958);
[4† ] L.Landau, E.Lifshits, Quantum Mechanics - non-relativistic theory. 3rd ed., Oxford, Pergamon Press, (1977);
[5† ] L.de Broglie, Heisenberg Uncertainties and Probabilistic Interpretation of Wave
Mechanics, Dordrecht, Kluwer Academic Publishers (1995) Chapter 8, eqs.(3) and (8);
[6† ] W.Price, S.Chissick, W.Heisenberg, The uncertainty principle and foundations of
quantum mechanics: a fifty years’ survey, New York, Wiley, (1977);
[7† ] A.Bohm, M.Loewe, Quantum mechanics: foundations and applications. 2nd rev.
and enl.ed., New York, NY, Springer-Verlag (1986);
[8† ] C.Cohen-Tannoudji, B.Diu, F.Laloe, Mecanique quantique, v.1, Paris, Hermann
(1973) ;
[9† ] L.Schiff, Quantum Mechanics, 3rd ed., International series in pure and applied
physics, New York, McGraw-Hill (1968);
9
[10† ] A. Messiah, Quantum Mechanics, v.1, North-Holland Publishing Company, Amsterdam (1961) p.300, eq.(VIII.9); actually the author had been very close to the general relation, only that he had not taken the centralized covariance, see the proof below
eq.(VIII.9);
[11† ] E.Merzbacher, Quantum Mechanics, 3rd ed.,New York, NY, Wiley (1998) p.219,
eq. (10.58); the same situation with the 1st and 2nd editions;
[12† ] A.Das, C.Melissinos, Quantum mechanics: a modern introduction, New York,
NY., Gordon & Breach, (1986);
[13† ] J.Sakurai, S.Tuan, Modern quantum mechanics. Rev. ed., Reading, MA, Addison - Wesley, (1994);
[14† ] F.Schröeck, Quantum mechanics on phase space, Dordrecht, Kluwer Academic
Publishers (1996);
[15† ] J.Perina, Z.Hradil, B.Jurco, Quantum optics and fundamentals of physics, Dordrecht, Kluwer Academic Publishers (1994);
[16† ] About Heisenberg uncertainty relation, translated in russian by A.Rogali, pp.210217, in “E.Schrödinger, Izbrannie trudi po kvantovoi mehanike (Collected papers on quantum mechanics)† , Moscow, Nauka (1976)”; the russian translation of the sentence (*) is:
“However, this never happens.”;
[17† ] G.Korn, T.Korn, Mathematical handbook for scientists and engineers, sec. enl.
and rev. ed., New York, N.Y., McGraw-Hill Publs. Co., (1968) p.604, eq.(18.4-10);
[18† ] H.Tucker, An introduction to probability and mathematical statistics, New York,
NY, Academic Press, (1963) p.91;
[19† ] C.Gardiner, Handbook of stochastic methods, 2nd ed., Berlin, Springer-Verlag,
(1985);
[20† ] A.Angelow, Physica A, 256 (1998) pp.485-498, the last equation in (24) and
equation (B.4).
10